To change the format slightly, with 東京__行きます。Which is better out of に and へ and why?

Well, へ is direction, に is destination. Assuming that 東京 as the destination is already fixed, then に is better. I know that に is better with past tense, but am not as sure for present/future tense. (Since the destination hasn't been reached yet, and thus might change, へ might be better.)

Yup. まい is an old form, still seen in some common phrases like that one. The meaning is negative-violition/intention/will (e.g. to have the intention of not doing something) or negative-probability (something will probably not happen, or will naturally not happen).